Symmetric 1 1 2 ‐Designs and 1 1 2 ‐Difference Sets

In this paper, we introduce a method to construct 1 1 2 ‐designs, which are also known as partial geometric designs, by using subsets of certain finite groups. We introduce the concept of 1 1 2 ‐difference sets and investigate the existence and nonexistence of these structures. We also provide some nonexistence results on 1 1 2 ‐designs based on the fact that 1 1 2 ‐designs yield directed strongly regular graphs.